Abstract
In this chapter the basic notions of lattice theory are collected. Besides the definition of a lattice the important definitions are the distributivity, the modularity and the orthomodularity of a lattice (Definitions 3.5, 3.6 and 3.10). The proposition stating that a lattice on which a finite dimension function exists is necessarily modular (Proposition 3.3) will become important in the Chapter 6. The one-to-one correspondence between prime filters in a lattice and lattice homomorphisms on the lattice into a Boolean lattice (Proposition 3.11) will be used later to show that there exist no lattice homomorphisms from a quantum logic into a Boolean lattice. The notion of a partial algebra, partial Boolean algebra and partial algebra homomorphism will come up in Chapter 9 naturally in connection with a certain concept of hidden variable theory of quantum mechanics.
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© 1998 Springer Science+Business Media Dordrecht
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Rédei, M. (1998). Lattice theoretic notions. In: Quantum Logic in Algebraic Approach. Fundamental Theories of Physics, vol 91. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9026-6_3
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DOI: https://doi.org/10.1007/978-94-015-9026-6_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4976-6
Online ISBN: 978-94-015-9026-6
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